Lecture 1: First-order ordinary differential equations.
Basics. First-order linear ordinary differential equations (ODEs). First-order non linear ODEs. Direction fields. Existence and uniqueness of solutions of the initial value problem (IVP). Numerical methods: Euler, Heun and Runge-Kutta.
Lecture 2: Second-order ordinary differential equations.
Linear second-order ODEs: Homogenous. Variation of paremeters and inhomogeneous ODEs. Method of undetermined coefficients. Linear ODEs with constant coefficients. Equidimensional ODEs (Euler-Cauchy). Reduction of order.
SUPPLEMENTARY MATERIAL: Linear oscillator and resonance.
Lecture 3: Systems of differential equations.
Systems of first-order ODEs: Linear systems. Autonomous linear systems. Two-dimensional homogeneous linear systems. Inhomogeneous linear systems. Variation of parameters. Undetermined coefficients.
SUPPLEMENTARY MATERIAL: Reduction to normal modes.
Lecture 4: Boundary value problems.
Existence of solutions of a boundary value problem (BVP). BVP for nonlinear ODEs.
SUPPLEMENTARY MATERIAL: Shooting method for linear BVPs.
Lecture 5: Fourier series and separation of variables: Heat equation.
Classification of second-order Partial Differential Equations (PDEs). Heat equation. Fourier series method for the heat equation: Separation of variables for the homogeneous heat equation. Eigenvalue problem. Homogeneous heat equation with Neumann boundary conditions and with periodic boundary conditions. Inhomogeneous problems. Properties of Fourier series.
SUPPLEMENTARY MATERIAL: Finite difference solution of the heat equation.
Lecture 6: Fourier series and separation of variables: Wave equation.
Derivation of the 1D wave equation. Boundary conditions. Fourier series method for the wave equation: Separation of variables for the homogeneous wave equation. Inhomogeneous wave equation. Resonance.
SUPPLEMENTARY MATERIAL: Finite difference solution of the wave equation.
Lecture 7: Fourier series and separation of variables: Laplace equation.
2D Laplace equation. Laplace equation for a rectangular region and for a disk. Cualitative properties of the Laplace equation.
SUPPLEMENTARY MATERIAL: Poisson equation.